227 research outputs found
Randomized Quantization and Source Coding with Constrained Output Distribution
This paper studies fixed-rate randomized vector quantization under the
constraint that the quantizer's output has a given fixed probability
distribution. A general representation of randomized quantizers that includes
the common models in the literature is introduced via appropriate mixtures of
joint probability measures on the product of the source and reproduction
alphabets. Using this representation and results from optimal transport theory,
the existence of an optimal (minimum distortion) randomized quantizer having a
given output distribution is shown under various conditions. For sources with
densities and the mean square distortion measure, it is shown that this optimum
can be attained by randomizing quantizers having convex codecells. For
stationary and memoryless source and output distributions a rate-distortion
theorem is proved, providing a single-letter expression for the optimum
distortion in the limit of large block-lengths.Comment: To appear in the IEEE Transactions on Information Theor
Vector Quantization with Error Uniformly Distributed over an Arbitrary Set
For uniform scalar quantization, the error distribution is approximately a
uniform distribution over an interval (which is also a 1-dimensional ball).
Nevertheless, for lattice vector quantization, the error distribution is
uniform not over a ball, but over the basic cell of the quantization lattice.
In this paper, we construct vector quantizers with periodic properties, where
the error is uniformly distributed over the n-ball, or any other prescribed
set. We then prove upper and lower bounds on the entropy of the quantized
signals. We also discuss how our construction can be applied to give a
randomized quantization scheme with a nonuniform error distribution.Comment: 22 pages, 3 figures. Short version presented at 2023 IEEE
International Symposium on Information Theor
A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon's Needle
In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of
geometric probability theory by defining an enlightening problem: What is the
probability that a needle thrown randomly on a ground made of equispaced
parallel strips lies on two of them? In this work, we show that the solution to
this problem, and its generalization to dimensions, allows us to discover a
quantized form of the Johnson-Lindenstrauss (JL) Lemma, i.e., one that combines
a linear dimensionality reduction procedure with a uniform quantization of
precision . In particular, given a finite set of points and a distortion level , as soon as , we can (randomly) construct a mapping from
to that approximately
preserves the pairwise distances between the points of .
Interestingly, compared to the common JL Lemma, the mapping is quasi-isometric
and we observe both an additive and a multiplicative distortions on the
embedded distances. These two distortions, however, decay as when increases. Moreover, for coarse quantization, i.e., for high
compared to the set radius, the distortion is mainly additive, while
for small we tend to a Lipschitz isometric embedding. Finally, we
prove the existence of a "nearly" quasi-isometric embedding of into . This one involves a non-linear
distortion of the -distance in that vanishes for distant
points in this set. Noticeably, the additive distortion in this case is slower,
and decays as .Comment: 27 pages, 2 figures (note: this version corrects a few typos in the
abstract
Digital Color Imaging
This paper surveys current technology and research in the area of digital
color imaging. In order to establish the background and lay down terminology,
fundamental concepts of color perception and measurement are first presented
us-ing vector-space notation and terminology. Present-day color recording and
reproduction systems are reviewed along with the common mathematical models
used for representing these devices. Algorithms for processing color images for
display and communication are surveyed, and a forecast of research trends is
attempted. An extensive bibliography is provided
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